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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{多元统计分析练习3.1-3.2}
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\date{2024 年 3 月 19 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设随机向量 $u=(u_1,u_2,\cdots,u_p)'$, 其中 $u_1,u_2,\cdots,u_p$ 独立同分布于 $N(0,1)$. 
设一个非退化线性变换 $x=\mu+Au$, 其中 $A$ 是一个 $p$ 阶非退化线性变换。
求随机向量 $x$ 的概率密度函数。

\vspace{0.2cm}

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\item  %Problem 02
写出二元正态分布 $x\sim N_2(\mu,\Sigma)$ 的密度函数和边缘密度函数，画出密度函数的等高线，解释相关系数 $\rho$ 和广义方差 $|\Sigma|$ 在密度函数的图像上的含义。

\vspace{0.2cm}

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\item  %Problem 03
在二元正态分布  $x\sim N_2(\mu,\Sigma)$ 中，设相关系数 $\rho=\pm 1$, 证明此时协方差矩阵 $\Sigma$ 不存在逆阵。此时称 $x$ 服从退化的二元正态分布。举出一个例子，说明退化的三元正态分布。

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\item  %Problem 04
多元正态分布 $x\sim N_p(\mu, \Sigma)$ 的特征函数定义为
$$\varphi_x(t) = \exp\left( it'\mu - \frac{1}{2}t'\Sigma t \right), $$
其中 $t=(t_1,\cdots,t_p)'$ 是一个 $p$ 维的未知数列向量。写出二元正态分布的特征函数。

\vspace{0.2cm}

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\item  %Problem 05
设 $x=(x_1,x_2)' \sim N_2(\mu, \Sigma)$,  求 $x_1-x_2$ 的分布。
一般地，设 $a=(a_1,a_2)'$ 是常数向量，求 $a'x$ 的分布。

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\item  %Problem 06
设 $x=(x_1,x_2,x_3,x_4)' \sim N_4(\mu, \Sigma)$,  求 $x_i$ 的分布。求 $(x_1,x_4)'$ 的分布。
求 $(x_4,x_1,x_3)'$ 的分布。

\vspace{0.2cm}

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\item  %Problem 07
设 $x\sim N_3(\mu,\Sigma)$, 其中 
$$
\Sigma=\begin{pmatrix} 3&0&0 \\ 0&5&-1 \\ 0&-1&1 \end{pmatrix}. 
$$
证明 $x_2$ 和 $x_3$ 不独立，$x_1$ 和 $(x_2,x_3)$ 独立。

\vspace{0.2cm}

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\item  %Problem 08
设 $x\sim N_p(\mu,\Sigma)$, 其中 $\Sigma$ 正定。
取 $c^2=\chi^2_\alpha(p)$, 则 $x$ 落在集合 
$$\{x: \,\, (x-\mu)'\Sigma^{-1}(x-\mu)\le c^2\}$$
的概率为 $1-\alpha$. 

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\item  %Problem 09
设 $x\sim N_2(\mu,\Sigma)$, 这里 
$$
x=\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}, \,\,
\mu=\begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \,\,
\Sigma=\begin{pmatrix} \sigma_{1}^2 & \sigma_{1}\sigma_{2}\rho \\ \sigma_1\sigma_{2}\rho & \sigma_{2}^2 \end{pmatrix}. 
$$
求 $x_2$ 对 $x_1$ 的线性回归函数 $E(x_1\mid x_2)$. 

\vspace{0.2cm}

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\item  %Problem 10
设 $x\sim N_2(\mu,\Sigma)$, 这里 
$$
\mu=\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}, \,\,
\Sigma=\begin{pmatrix} 16 & -4 & 2 \\ -4 & 4 & -1 \\ 2 & -1 & 4 \end{pmatrix}. 
$$
求已知 $x_1+2x_3$ 时，$(x_2-x_3, x_1)'$ 的条件分布。

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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